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<b>let </b><font color="Maroon">C</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">F1</font> be  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">F2</font> be  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">G1</font> be  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">t</font> be    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font>;<br/>













<b>assume </b><a NAME="E1:45"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">F1</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 ;<br/>

<a NAME="E2:45"/>
<font color="Maroon">G1</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="functor2.html#T9">FUNCTOR2:9</a></i>;<br/>
<b>hence </b><a NAME="E3:45"/>
<font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F2</font>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D7">FUNCTOR2:def 7</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E4:45"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>

<a NAME="E5:45"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F1</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, <a class="ref" href="functor2.html#D6">FUNCTOR2:def 6</a></i>;<br/>
<a NAME="E6:45"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<a NAME="E7:45"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<a NAME="E8:45"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F2</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<a NAME="E9:45"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F2</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<a NAME="E10:45"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/>
<a NAME="E11:45"/><i><font color="Green">E104</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/>
<a NAME="E12:45"/><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/>
<a NAME="E13:45"/><i><font color="Green">E106</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G1tb</font> = <font color="Maroon">G1</font> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="functor0.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G1F1f</font> = <font color="Maroon">G1</font> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font></span>)</span> as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <b>by </b><i>, <a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G1ta</font> = <font color="Maroon">G1</font> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span> as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="functor0.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F2</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t</font></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font></span>)</span> = 
<font color="Maroon">G1tb</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">G1tb</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">G1F1f</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">G1</font> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">b</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="txt" href="functor3.html#E4:45"><i><font color="Green">E32</font></i></a>, <a class="ref" href="functor0.html#D24">FUNCTOR0:def 24</a></i>
<br/>.= 
<font color="Maroon">G1</font> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="functor2.html#D7">FUNCTOR2:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="functor0.html#K6">.</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="functor0.html#K6">.</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E1:45"><i><font color="Green">E31</font></i></a>, <a class="txt" href="functor3.html#E5:45"><i><font color="Green">E33</font></i></a>, <a class="ref" href="functor0.html#D24">FUNCTOR0:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F2</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">G1ta</font>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F2</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t</font></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , </i>
;<br/>


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